The history of higher mathematics is a piling on of levels of abstraction as
the search for greater generalization goes on. In effect, arithmetic was created
as a generalization of the principles of counting, summarizing in rules what
the result would be of various kinds of operations (adding things together,
removing some, partitioning, recombining). Algebra gives higher-order gener-
alizations about whole classes of operations that might be carried out in
arithmetic (thus it is always possible to check an algebraic equation with a
specific numerical example). The takeoff of modern mathematics around the
time of Tartaglia and Cardano was in formulating higher-order principles as
to how to solve whole classes of equations; these in turn gave rise to still further
questions as to whether general solutions were possible in various areas (e.g.,
the famous question of the general solution of the quintic equation, a flashpoint
for abstract mathematics in the 1820s). The revolution of higher mathematics
was the formulation of still higher levels of abstraction for dealing with these
meta-issues, such as the theory of groups. This turned revolutionary as it
dawned on mathematicians that they were in the business of exploring succes-
sive levels of abstraction, and that they could create abstract systems at will
in order to solve their problems, or indeed just for the free creativity of
exploring these realms. The non-Euclidean geometries took off when it was
recognized that there was no longer any need to connect them with plausible
physical representations; at the same time, alternative algebras were invented
by deliberately varying the axiomatic sets of operations which underlie con-
ventional algebra.
The increasing reflexivity that went along with this movement had a double
direction: piling on still higher levels of abstraction, and also delving back-
wards into the concepts which were used in a taken-for-granted, non-reflexive
manner earlier in the history of mathematics. Higher algebra was created when
it was realized that the set of axioms underlying elementary algebra were only
one of the possible abstract combinations possible. Plunging still further back,
one arrives at arithmetic, and then at the concept of number itself. Here the
conservative mathematics drew the line, Kronecker declaring, “God made the
integers, man created all the rest!” and blasting the more extreme forms of
revolutionary mathematicians for its monstrosities and paradoxes. Neverthe-
less, this terrain too was aggressively opened up, by Cantor and Frege. Ex-
traplating the path of previous revolutionary reconstructions, Frege recognized
that numbers are not primitive things, but emerge in systems of operations,
and indeed there are multiple dimensions of these operations. In the same way
that Cantor distinguished several orders of infinity by several methods of
counting, Frege distinguished cardinal and ordinal properties of numbers.
More generally, by examining the operations that are implied in using the equal
sign () in an equation, he uncovered multiple dimensions in our taken-for-
granted concept of equality; with greater generality this became the distinction
Sequence and Branch in the Social Production of Ideas^ •^853